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Modified Stencils for Boundaries and Subgrid Scales in the Finite-Difference Time-Domain Method
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30).
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis centers on modified stencils for the Finite-Difference Time-Domain method (FDTD), or Yee scheme, when modelling curved boundaries, obstacles and holes smaller than the discretization length.  The goal is to increase the accuracy while keeping the structure of the standard method, enabling improvements to existing implementations with minimal effort.

We present an extension of a previously developed technique for consistent boundary approximation in the Yee scheme.  We consider both Maxwell's equations and the acoustic equations in three dimensions, which require separate treatment, unlike in two dimensions.

The stability properties of coefficient modifications are essential for practical usability.  We present an analysis of the requirements for time-stable modifications, which we use to construct a simple and effective method for boundary approximations. The method starts from a predetermined staircase discretization of the boundary, requiring no further data on the underlying geometry that is being approximated.

Not only is the standard staircasing of curved boundaries a poor approximation, it is inconsistent, giving rise to errors that do not disappear in the limit of small grid lengths. We analyze the standard staircase approximation by deriving exact solutions of the difference equations, including the staircase boundary. This facilitates a detailed error analysis, showing how staircasing affects amplitude, phase, frequency and attenuation of waves.

To model obstacles and holes of smaller size than the grid length, we develop a numerical subgrid method based on locally modified stencils, where a highly resolved micro problem is used to generate effective coefficients for the Yee scheme at the macro scale.

The implementations and analysis of the developed methods are validated through systematic numerical tests.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2012. , xi, 34 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2012:07
Keyword [en]
FDTD, Yee, Staircasing
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-95510ISBN: 978-91-7501-417-3 (print)OAI: oai:DiVA.org:kth-95510DiVA: diva2:528722
Public defence
2012-06-15, F3, Lindstedtsvägen 26, KTH, Stockholm, 10:00 (English)
Opponent
Supervisors
Funder
Swedish e‐Science Research Center
Note

QC 20120530

Available from: 2012-05-30 Created: 2012-05-28 Last updated: 2013-04-09Bibliographically approved
List of papers
1. On Energy Preserving Consistent Boundary Conditions for the Yee Scheme in 2D
Open this publication in new window or tab >>On Energy Preserving Consistent Boundary Conditions for the Yee Scheme in 2D
2012 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 52, no 3, 615-637 p.Article in journal (Refereed) Published
Abstract [en]

The Yee scheme is one of the most popular methods for electromagnetic wave propagation. A main advantage is the structured staggered grid, making it simple and efficient on modern computer architectures. A downside to this is the difficulty in approximating oblique boundaries, having to resort to staircase approximations. In this paper we present a method to improve the boundary treatment in two dimensions by, starting from a staircase approximation, modifying the coefficients of the update stencil so that we can obtain a consistent approximation while preserving the energy conservation, structure and the optimal CFL-condition of the original Yee scheme. We prove this in L_2 and verify it by numerical experiments.

Place, publisher, year, edition, pages
Springer, 2012
Keyword
Yee scheme, FDTD, Computational electromagnetics
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-73240 (URN)10.1007/s10543-012-0376-2 (DOI)000308234600006 ()2-s2.0-84865704007 (Scopus ID)
Funder
Swedish e‐Science Research Center
Note

QC 20120919

Available from: 2012-02-02 Created: 2012-02-02 Last updated: 2017-12-08Bibliographically approved
2. Accuracy of staircase approximations in finite-difference methods for wave propagation
Open this publication in new window or tab >>Accuracy of staircase approximations in finite-difference methods for wave propagation
2014 (English)In: Numerische Mathematik, ISSN 0029-599X, E-ISSN 0945-3245, Vol. 128, no 4, 741-771 p.Article in journal (Refereed) Published
Abstract [en]

While a number of increasingly sophisticated numerical methods have been developed for time-dependent problems in electromagnetics, the Yee scheme is still widely used in the applied fields, mainly due to its simplicity and computational efficiency. A fundamental drawback of the method is the use of staircase boundary approximations, giving inconsistent results. Usually experience of numerical experiments provides guidance of the impact of these errors on the final simulation result. In this paper, we derive exact discrete solutions to the Yee scheme close to the staircase approximated boundary, enabling a detailed theoretical study of the amplitude, phase and frequency errors created. Furthermore, we show how evanescent waves of amplitude occur along the boundary. These characterize the inconsistencies observed in electromagnetic simulations and the locality of the waves explain why, in practice, the Yee scheme works as well as it does. The analysis is supported by detailed proofs and numerical examples.

Keyword
FDTD, Yee, Staircasing
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-95502 (URN)10.1007/s00211-014-0625-1 (DOI)000344752100005 ()
Funder
Swedish e‐Science Research Center
Note

QC 20141212

Updated from manuscript to article in journal.

Available from: 2012-05-28 Created: 2012-05-28 Last updated: 2017-12-07Bibliographically approved
3. Consistent modeling of boundaries in acoustic finite-difference time-domain simulations
Open this publication in new window or tab >>Consistent modeling of boundaries in acoustic finite-difference time-domain simulations
2012 (English)In: Journal of the Acoustical Society of America, ISSN 0001-4966, E-ISSN 1520-8524, Vol. 132, no 3, 1303-1310 p.Article in journal (Refereed) Published
Abstract [en]

The finite-difference time-domain method is one of the most popular for wave propagation in the time domain. One of its advantages is the use of a structured staggered grid, which makes it simple and efficient on modern computer architectures. A drawback however is the difficulty in approximating oblique boundaries, having to resort to staircase approximations.  In many scattering problems this means that the grid resolution required to obtain an accurate solution is much higher than what is dictated by propagation in a homogeneous material.  In this paper zero boundary data is considered, first for the velocity and then the pressure. These two forms of boundary conditions model perfectly rigid and pressure-release boundaries, respectively.  A simple and efficient method to consistently model curved rigid boundaries in two dimensions was developed in [A.-K. Tornberg and B. Engquist, J. Comput. Phys. 227, 6922--6943 (2008)].  Here this treatment is generalized to three dimensions.  Based on the approach of this method, a technique to model pressure-release surfaces with second order accuracy and without additional restriction on the timestep is also introduced.  The structure of the standard method is preserved, making it easy to use in existing solvers.  The effectiveness is demonstrated in several numerical tests.

Place, publisher, year, edition, pages
Acoustical Society of America (ASA), 2012
Keyword
Computer architecture, Finite difference time domain method
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-63593 (URN)10.1121/1.4740490 (DOI)000309155000021 ()2-s2.0-84866306609 (Scopus ID)
Funder
Swedish e‐Science Research Center
Note

QC 20121031

Available from: 2012-01-23 Created: 2012-01-23 Last updated: 2017-12-08Bibliographically approved
4. On Consistent Boundary Conditions for the Yee Scheme in 3D
Open this publication in new window or tab >>On Consistent Boundary Conditions for the Yee Scheme in 3D
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The standard staircase approximation of curved boundaries in the Yee scheme is inconsistent. Consistency can however be achieved by modifying the algorithm close to the boundary.  We consider a technique to consistently model curved boundaries where the coefficients of the update stencil is modified, thus preserving the Yee structure.  The method has previously been successfully applied to acoustics in two and three dimension, as well as electromagnetics in two dimensions.  In this paper we generalize to electromagnetics in three dimensions.  Unlike in previous cases there is a non-zero divergence growth along the boundary that needs to be projected away.  We study the convergence and provide numerical examples that demonstrates the improved accuracy.

Keyword
FDTD, Yee, Staircasing
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-95504 (URN)
Note
QC 20120530Available from: 2012-07-30 Created: 2012-05-28 Last updated: 2012-07-30Bibliographically approved
5. Numerical subgrid scale models for the Yee scheme
Open this publication in new window or tab >>Numerical subgrid scale models for the Yee scheme
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The Yee scheme is a very common and practical algorithm for the simulation of wave propagation on uniform grids.  We develop numerical subgrid scale models in order to incorporate effects of obstacles and holes that are smaller than the grid spacing. The models are based on pre-computing at the microscale, and are thus including the effect of the detailed small scale shape.  Numerical examples in 1D, 2D and 3D are given.

Keyword
Yee, FDTD, Subcell
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-95505 (URN)
Note
QS 2012Available from: 2012-07-30 Created: 2012-05-28 Last updated: 2012-07-30Bibliographically approved

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